Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/254

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EXAMPLES[1]

1. Expand in powers of . Ans.
2. Expand in powers of . Ans.
3. Expand in powers of . Ans.
4. Expand in powers of . Ans.
5. Expand in powers of .
6. Expand in powers of .
7. Expand in powers of .
8. Expand in powers of .
9. Expand in powers of .
10. Expand in powers of . Ans.
11. Expand in powers of . Ans.
12. Expand in powers of . Ans.
13. Expand the following in in powers of .
(a)
(b)

145. Maclaurin's Theorem and Maclaurin's Series. A particular case of Taylor's Theorem is found by placing in (61), §144, giving

(64)

where lies between 0 and . (64) is called Maclaurin's Theorem. The right-hand member is evidently a series in in the same sense that (62), §144, is a series in .

Placing in (62), §144 we get Maclaurin's Series'[2],

(65)
  1. In these examples assume that the functions can be developed into a power series.
  2. Named after [Maclaurin] (1698-1746), being first published in his [Treatise of Fluxions], Edinburgh, 1742. The series is really due to [[1]] (1692-1770).